I/O and I/S/O Relationships in Time and Frequency 2.5 2002/02/15 2008/01/30 17:26:39.444 US/Central Elizabeth Chan lizychan@rice.edu Elizabeth Chan lizychan@rice.edu (Blank Abstract)

I/O and I/S/O representation of SISO linear systems I/O I/S/O variables: ( u , y ) variables: ( u , x , y ) t q y t t p u t , n deg q deg p t xt A x t B u t , y t C x t D u t u t , y t x t n , A B C D n 1 n 1 Impulse Response t q h t t p δt h t D δ t C A t B , t 0 H s ℒ h t p s q s H s D C s I A B Poles - characteristic roots - eigenfrequencies λi , q λi 0 , I 1 , … , n λi I A 0 Zeros H zi 0 ⇔ p zi , 1 , … , n zi I A -B -C -D 0 Matrix exponential A t k 0 t k k A k t At A A t A t A ℒ A t s I A BIBO stability y h u , requirement u Norm u u ⇔ h 1 t 0 h t ⇔ λi 0 ⇔ poles LHP Solution in the time domain y t yzi t yzs t x t xzi t xzs t y t I 1 n ci λi t τ 0- t h t τ u τ x t A t x 0- τ 0- t A t τ B u τ y t C A t x 0- τ 0- t D δ t τ C A t τ B u τ , h · D δ t τ C A t τ B y t C A t x 0- τ 0- t h t τ u τ Laplace Transform: Solution in the frequency domain Y s r s q s H s U s X s s I A x 0- s I A B U s Y s C s I A x 0- D C s I A B U s , H s D C s I A B

Definition of state from I/O description Let H s D p s q s , deg p deg q . Define w so that t q w t u t , y t t p w D u t xT w w1 … wn1 n , n : degree of q s .

Various Responses Zero-input or free response response due to initial conditions alone. Zero-state or forced response response due to input (forcing function) alone (zero initial condition). Homogeneous solution general form of free-response (arbitrary initial conditions). Particular solution forced response. Steady-state response response obtained for large balues of time T . Transient response full response minus steady minus state response.